D'Alembertian solutions of linear differential and difference equations

ABSTRACT

D'Alembertian solutions of differential (resp. difference) equations are those expressible as nested indefinite integrals (resp. sums) of hyperexponential functions. They are a subclass of Liouvillian solutions, and can be constructed by recursively finding hyperexponential solutions and reducing the order. Knowing d'Alembertian solutions of Ly = 0, one can write down the corresponding solutions of Ly = f and of L*y = 0.

REFERENCES

• Abr89: S. A. Abramov, Rational solutions of linear differential and difference equations with polynomial coefficients, USSR Computational Mathematics and Mathematical Physics, v.29 n.6, p.7-12, 1989 [doi>10.1016/S0041-5553(89)80002-3]
• Abr&Kva91: S. A. Abramov , K. Yu. Kvansenko, Fast algorithms to search for the rational solutions of linear differential equations with polynomial coefficients, Proceedings of the 1991 international symposium on Symbolic and algebraic computation, p.267-270, July 15-17, 1991, Bonn, West Germany [doi>10.1145/120694.120735]
• Abr93: S. A. Abramov, On d'Alembert substitution, Proceedings of the 1993 international symposium on Symbolic and algebraic computation, p.20-26, July 06-08, 1993, Kiev, Ukraine [doi>10.1145/164081.164086]
• Abr94a: S. A. Abramov, D'Alembert substitutions and adjoint differential equations (computer-algebraic aspects), Computational Mathematics and Mathematical Physics, v.34 n.7, p.863-873, Nov. 1994
• Abr94b: S.A. Abramov (1994): D'Alembertian solutions of nonhomogeneous linear ordinary differential and difference equations. To be published.
• Alm&Zei90: G. Almkvist , D. Zeilberger, The method of differentiating under the integral sign, Journal of Symbolic Computation, v.10 n.6, p.571-591, Dec. 1990 [doi>10.1016/S0747-7171(08)80159-9]
• Bro92a: Manuel Bronstein, On solutions of linear ordinary differential equations in their coefficient field, Journal of Symbolic Computation, v.13 n.4, p.413-439, April 1992 [doi>10.1016/S0747-7171(08)80106-X]
• Bro92b: Manuel Bronstein, Linear ordinary differential equations: breaking through the order 2 barrier, Papers from the international symposium on Symbolic and algebraic computation, p.42-48, July 27-29, 1992, Berkeley, California, United States [doi>10.1145/143242.143264]
• Bro&Pet94: M. Bronstein, M. Petkov~ek (1994): On Ore rings, linear operators and factorisation, Programmirovanie 1, no. 1. To appear.
• Gos78: R.W. Gosper, Jr. (1978): Decision procedure for indefinite hypergeometric summation, Proc. Natl. A cad. Sci. USA 75, 40-42.
• Koo91: R.J. Kooman (1991): Convergence properties of recurrence sequence, Centrum voor Wiskunde en Informatica, CWI Tract 83, Amsterdam.
• Ore33: O. Ore (1933): Theory of non-commutative polynomials, Ann. Maths. 34, 480-508.
• Pet92: Marko Petkovšek, Hypergeometric solutions of linear recurrences with polynomial coefficients, Journal of Symbolic Computation, v.14 n.2-3, p.243-264, Aug./Sept. 1992 [doi>10.1016/0747-7171(92)90038-6]
• Sin81: M.F. Singer (1981): Liouvillian solutions of n-th order homogeneous linear differential equations, Amer. J. Math. 103, 661-681.
• Sin91: Michael F. Singer, Liouvillian solutions of linear differential equations with Liouvillian coefficients, Journal of Symbolic Computation, v.11 n.3, p.251-273, March 1991 [doi>10.1016/S0747-7171(08)80048-X]

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Algebraic algorithms

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on polynomials

  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Nonalgebraic algorithms

General Terms:
Algorithms, Theory, Verification